函数极限问题设极限limx→1f(x)存在,且f(x)=(x^2+4x-5)/(x^2-1)+sin(2-2x)/(x-1)乘以limx→1f(x),求limx→1f(x)（sin(2-2x)/(x-1)）乘以limx→1f(x)

Question

函数极限问题
设极限lim x→1 f(x)存在,且f(x)=(x^2+4x-5)/(x^2-1)+sin(2-2x)/(x-1)乘以limx→1 f(x),求lim x→1 f(x)
（sin(2-2x)/(x-1)）乘以limx→1 f(x)

Answer 1

记lim[x→1] f(x)=a
f(x)=(x²+4x-5)/(x²-1)+asin(2-2x)/(x-1)
两边取极限得：
lim[x→1] f(x)=lim[x→1] [(x²+4x-5)/(x²-1)+asin(2-2x)/(x-1)]
=lim[x→1] [(x-1)(x+5)/(x²-1)+a(2-2x)/(x-1)]
=lim[x→1] [(x+5)/(x+1)-2a]
=3-2a
则：a=3-2a,解得：a=1
因此：lim[x→1] f(x)=1
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